The Riemann hypothesis: the English mathematician, claims to have solved the problem of the Millennium

SL, Вторник 25 Сентябрь 2018 - 15:58:39

The famous scientist Michael Atiyah, winner of the fields medal award, gave a proof of the Riemann hypothesis. This issue is included in the list of the six greatest unsolved problems. If successful, the researcher will get a million dollars.

"Solve the Riemann hypothesis, and you will become famous. If you're already famous, you become infamous, says Atiyah material in the New Scientist. during a conversation. – No one believes any proof of the Riemann hypothesis, because this task is so difficult. No one has proven it, so why would anyone prove it now? Of course, if you have no new ideas".

To explain what is the Riemann hypothesis will have to start from afar. Recall that an integer is called Prime if no remainder is divisible only by itself and by one (but is not itself a unit). For example, 2, 3, 5, 7, 11 – Prime numbers and 4, 6, 8, 9 – no.

Prime numbers have long attracted the attention of mathematicians. The fact that any integer is either itself simple, or it may be derived from the simple by their multiplication. For example, 8 = 2×2×2 and 9 = 3×3. Thus, primes are the "building blocks" that comprise all integers with the operation multiplication.


 

Is this a mathematical phenomenon and the practical side. Multiply a few simple numbers – a much more easy task than to produce the reverse process: take the result and calculate the product of any Prime numbers it is. The enumeration of all possible candidates is too long. Everyone can verify this from experience, trying to figure out using a calculator, what is the Prime factorization decomposed, say, 1059811.

If the Prime factors are large enough, then find them on the product – a task daunting even for modern computers. And in this construct the encryption algorithms that protect, say, our financial accounts. The key consists of several very large primes. The computer multiplies them. If the artwork converges with that which is stored in the database, given access to information. Knowing only the product, find its factors (to pick up the key) it is impossible for any reasonable period of time.

Our accounts protects the fact that is not known (and may not exist) no fast algorithm to find all primes from one to a given number x. However, if we cannot know what kind of numbers, can we at least find out how many of them?

A function that specifies the number of primes from one to a given number x is called the distribution function of Prime numbers. For any convenient formula is also not found, however, mathematicians with interest will investigate its properties.


In 1859, the great mathematician Bernhard Riemann made a hypothesis which was later named after him. If this hypothesis is correct, that is true and a lot of interesting statements about the distribution of Prime numbers.

As it is formulated? Here we have a little touch of mathematics. All the numbers that are studied in the school curriculum – positive, negative and zero, integers and fractional, rational and irrational mathematicians United under the name valid. However, there are also complex numbers.

This number represents the sum of a + i×b, where a, b are real numbers, i is the so-called imaginary unit. It is defined by the formula i2 = -1. If b = 0 then the complex number is just a real number a. Thus, the actual number is only a special case of complex.

It is clear that no real number squared cannot be negative. So mathematicians need the imaginary unit and complex numbers. They can solve the problem that, when operation of only real numbers appear simply meaningless. But with the help of this tool the following tasks are often elegantly solved, and these decisions, by the way, find many applications in physics and engineering. So that humanity has managed to invent, say, computers, is the "merit" of complex numbers.

Riemann came up with a function that today is called the Zeta Riemann function ζ(x). For its determination we with the permission of the reader to not give: it is cumbersome and requires the introduction of several mathematical concepts. We mention only two facts. First, this function is closely linked with the distribution of Prime numbers. Second, her argument is a complex number.


By definition the Zeta function, if its argument is a negative even number (recall that integers are too complex), then ζ(x) = 0. In other words, ζ(-2) = ζ(-4) = ζ(-6) = ... = 0. The question is, what other numbers this function vanishes.

The Riemann hypothesis is: if ζ(a + ib) = 0 and a + ib is not a negative even (described above), then a = ½.

This at first glance a simple statement trying to prove many great mathematicians, but nobody did. Mathematical Institute named clay (Clay Mathematics Institute) incorporated this hypothesis the seven Millennium problems for the solution of which was a reward of one million U.S. dollars. Note that while this list solved only one task. This is the poincaré conjecture, proved by Russian mathematician Grigory Perelman.

Atiyah argues that he has proved the Riemann hypothesis. Preprint evidence published in PDF file is five pages. According to New Scientist, it is based on the results of the great mathematicians of the twentieth century, John von Neumann and Friedrich Hirzebruch. Using their theorems, Atiyah proved the Riemann hypothesis by contradiction: he assumed that she was wrong, and came to a contradiction.

"That's wonderful, says Atiyah – but I say that all the hard work was done 70 years ago."

However, mathematics is, of course, need to be overhauled. No one is immune from error in such a difficult case as a mathematical proof, especially when we are talking about the Millennium problems. Attempts to prove the Riemann hypothesis, different authors have been published, but each time was incorrect. Maybe this time humanity will finally get lucky.

Recall that "Conduct.Science" (nauka.vesti.ru) previously wrote about other prominent mathematical works, including those made by Russian scientists.




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